Vector Projection

A projection of a vector u onto v is written as proj_v u. It is named as the vector projection of u onto v.

To find proj_v u, we use the formula below:

Take note that


where the scalar component of u in the direction of v is and the direction is . The scalar component tells us about the magnitude of the vector proj_v u and whether it's in the same direction of v or opposite to it. When the scalar component is positive, proj_v u is in the same direction with v; when it is negative, proj_v u is opposite to the direction of v.

Try to move point A below to the opposite direction of the blue vector, v. You will see that proj_v u is opposite to the direction of v.


Betty, Created with GeoGebra and Thomas Calculus, Pearson.

The Dot Product

Given two vectors u = < u₁, u₂, u₃ > and v = < v₁, v₂, v₃ >, and the angle between them is . The dot product between these two vectors are defined as below.

1. 
2. 

We can use either one to find the dot product, depending on the available information.

From the second definition, we can modify the equation to become


in order to find the angle between the two vectors.

Now, move the points A and B below and observe the changes of the calculation. The calculation on top of the graph is the calculation based on the first definition above. The bottom calculation is based on the second definition. Get yourself a calculator and calculate for both. You will find that both gives to the same answer.

Take note that and .


From the second definition, we can deduce the following statement

because .


Betty, Created with GeoGebra and Thomas Calculus, Pearson.